PASSENGE R PIGEON

Smithsonian Institution
Libraries

Given in memory of

Elisha Hanson
1

by
Letitia Armistead Hanson

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VOL. XII, NO. 5 | my | MAY, 1921
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THE SCIENTIFIC
MONTHLY

EDITED BY J. McKEEN CATTELL

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THE HISTORY OF MATHEMATICS. Professor Ernest W. Brown _____________ 385

DEMOCRATS AND ARISTOCRATS IN SCIENTIFIC RESEARCH. Dr. Esmond
1 Ag pac se RIM SSA TALE at RAVAN pe PI Ap ol A 414

THE CONDUCTION OF RESEARCH. F. H. Norton

A BIOLOGICAL EXAMINATION OF LAKE GEORGE. Professor James G.
Needham

THE HISTORY OF SCIENCE AS AN ERROR BREEDER. Professor G. A. Miller__439

THE BIOLOGY OF DEATH—THE CHANCES OF DEATH.
FrcOLesseraiwa yarn on cle treet rs Lae ei i930 a NR OU UI a 444

PUBLISHED FIGURES AND PLATES OF THE EXTINCT PASSENGER PIGEON.
Dr. R. W. Shufeldt

THE PROGRESS OF SCIENCE:

Professor Einstein’s Visit to the United States; The Massachusetts Institute of

Technology and President Nichols; The Service for Scientific News; Scientific
Items 82

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Entered ag second-class matter February 8, 1921, at the Post Office at Utica, N. Y., under the Act of March 3, 1879.

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THE SCIENTIFIC
MONTHLY

MAY, 1921

THE HISTORY OF MATHEMATICS*

By Professor ERNEST W. BROWN
YALE UNIVERSITY

HE earliest dawn of science is without doubt not different from
. that of intelligence. But the civilized man of to-day, far removed
as he is from the lowest of existing human races, is probably as far
again from the being whom one would not differentiate from the
animals as far as mental powers are concerned. What this difference
is, neither ethnologist nor psychologist can yet tell. Perhaps the
nearest approach to a definition, at least from the point of view of this
article, is contained in the distinction between unconscious and con-
scious observation. We are familiar with both sides even in ourselves:
records can be impressed on the brain and remain there apparently
dormant until some stimulus brings them to fruition, and again, record
and stimulus can appear together so that a train of thought is im-
mediately started.

The faculty of conscious observation is a fundamental requirement
of a scientific training, and no development can take place until it has
been acquired to some extent. Although this is only the first step, it
was probably a long one in the history of the race, just as it is relatively
long in the lives of the majority of individuals. Reasoning concern-
ing the observation follows, but not much success can be attained until
a considerable number of observations have been accumulated. We
may indeed put two and two together to make four, but experience
shows that with phenomena the answer is more often wrong than right:
we need much more information in order to get a correct answer.

We expect, then, that the earlier stage of a science will be one not
so much of discovery as of conscious observation of phenomena which
are apparent as soon as attention is called to them. The habit once
acquired, the search for the less obvious facts of nature begins, and

*A lecture delivered at Yale University, February 26, 1920, the first of a
series on the History of Science under the auspices of the Gamma Alpha
Graduate Scientific Fraternity.

VOL. XII.—25.

386 THE SCIENTIFIC MONTHLY

in the search many unexpected secrets are found. Once a body of facts
has been accumulated, correlation follows. The attempt is made to
find something common to them all and it is at this stage that science,
in the modern sense, may perhaps be said to have its birth. But this is
only a beginning. The mind that can grasp correlation can soon pro-
ceed to go further and try to find a formula which will not only be a
common property, but which will completely embrace the facts; that
is, in modern parlance, a law which groups all the phenomena under
one head.

The formula or law once discovered, consequences other than those
known are sought, and the process of scientific discovery begins.
Realms which could never have been opened up by observation alone
are revealed to the mind which has the ability to predict results as con-
sequences of the law, and thence is found the means by which the
truth of the law is tested. If the further consequences are shown to
be in agreement with what may be observed, the evidence is favorable.
If the contrary, the law must be abandoned or changed so as to embrace
the newly discovered phenomena. The process of trial and error, or
of hypothesis and test, is a recurring one which embraces a large pro-
portion of the scientific work of to-day.

Scientific development has two main aspects. One is the framing
of laws in order to discover new phenomena and develop the subject
forward so as to open out new roads into the vast forest of the secrets
of nature. The other is the turning backward in order to discover the
foundations on which the science rests. Just as no teacher would think
it wise to start the young pupil in chemistry or physics by introducing
him at the outset to the fundamental unit of matter or energy as it is
known at the time, but will rather start in the middle of the subject with
facts which are within the comprehension of his mind and the exper-
ience of his observation, science itself has been and must necessarily be
developed in the same manner. We proceed down to the foundations
as well as up to the phenomena.

This two-fold aspect of scientific research has had revolutionary
results in the experience of the last half century. It has fundamentally
changed the ideas of those who study the so-called laboratory subjects
in which observation with artificially constructed materials goes hand
in hand with the framing of laws and hypotheses, but it has changed
the study of mathematics in an even more fundamental manner. In the
past, geometry and arithmetic were suggested by observation and
practical needs and the development of both with symbolic representa-
tion proceeded on lines which were dictated by the problems which
arose. The methods of discovery in working forward were not es-
sentially different from those of an observational science, except per-
haps that the testing of a new law was unnecessary on account of the

THE HISTORY OF MATHEMATICS 387

rules of reasoning which accompanied them and became embodied into
a system of logic. It was, however, in proceeding backward to dis-
cover the foundations that the whole aspect of the subject changed.
While many of the hypotheses were suggested by observation and were
known by numerous tests to be applicable to the discussion of natural
phenomena, it became evident that the actual hypotheses used were in-
dependent of the phenomena. The laws which are at the basis of
geometrical reasoning are not necessarily natural laws: they can equally
well be regarded as mere productions of the brain in the same sense
as we might imagine a race of intelligent beings on another planet
free from some of our limitations or restricted by limitations from
which we are free. It is seen to be the same with the rules of symbolic
reasoning which have gradually grown up. A geometry without
Kuclid’s axiom of parallels has been constructed perfectly consistent
in all its parts. This is built up of a set of axioms which constitute its
foundation together with a code of reasoning by which we develop the
consequences of the axioms. The same is true of geometries which
involve space in more than three dimensions. It is somewhat easier to
imagine symbolic developments which have their foundations different
from those of our school and college algebras because there is no
obvious connection between these rules and the phenomena of nature.

It may be asked what limitation is there in the development of
mathematical theories if any set of axioms may be laid down.
Theoretically there is none, except that if we retain our code of reason-
ing about them such axioms must not be inconsistent with one another.
A certain sense of the fitness of things restrains mathematicians from
a wild overturning of the law and order which have been established
in the development of mathematics, just as it restrains democracies
from trying experiments in government which overturn too much the
existing order of affairs. Changes proceed in mathematics just as in
politics by evolution rather than by revolution. The slowly built up
structure of the past is not to be lightly overturned for the sake of
novelty.

The developments traced above apply mainly to the subject of
mathematics apart from its applications to the solution of problems
presented by nature. Applied mathematics is a method of reasoning
through symbols by which we can discover the consequences of the
assumed laws of nature. The symbolism which we adopt and the
rules we lay down with which to reason are immaterial, provided they
are convenient for the objects we have in view. One feature must not
be forgotten. We can never deduce the existence of phenomena through
mathematical processes which were not implicitly contained in the laws
of nature expressed at the outset in symbols. One cannot take out of
the mathematical mill any product which was not present in the raw

388 THE SCIENTIFIC MONTHLY

material fed into it. It is the purpose of the mill to work up the raw
material and the better the machinery the more finished and more
varied will be the product.

To write a connected story of mathematical development within the
limits of a brief article on a consistent plan without making it a mere
catalogue of names and results is a difficult, perhaps impossible, task
and no attempt is made here to accomplish it. If we try to lay stress
on the workers rather than on what they achieved, in one period we
encounter schools which developed particular subjects, in another,
outstanding figures with or without influence on contemporary de-
velopment and in still another numerous investigators who contributed
in varying degrees to the advances made in their age. On the
other hand if the development of ideas be the basis, parallel develop-
ments followed independently by different schools sometimes occur,
at another time general methods of treatment seem to pervade and
again we may find some fundamental advance made, the effect of
which is not felt for many years. Consequently the plan which seems
to fit best any particular period has been adopted for that period.
Another difficulty consists in assigning the relative values to either men
or ideas, about which probably no two persons will agree. There is,
however, one stumbling block which is peculiar to mathematics. The
very names themselves of many important branches of pure mathe-
matics convey no meaning to the majority of scientific readers who are
not trained mathematicians and to whom alone this article is intended
to appeal. An attempt at brief definition has sometimes been made,
but it can at best only give a partial view of the subject even with
concrete illustrations of its significations. In the general outline, the
historical development has been followed, but the methods of carrying
it forward have varied with different periods. When a choice of names
mentioned has to be made, a rough guide has been furnished in the
earlier periods by selecting those who have taken the step forward
which has rendered the subject capable of expansion or application
by others as judged in the light of present knowledge. In the nine-
teenth and present centuries to carry out this method has proved to be
beyond the ability of the writer; consequently, in most cases personal
mention has only been incidental and the names of many of those who
have done great service are missing. Fortunately, the history of mathe-
matics has received much attention in articles and separate volumes and
to them the reader who is interested in obtaining fuller information is
referred.

The earliest traces in the form of written records have come to us
from the Babylonians, mainly in the form of clay tablets which appear

THE HISTORY OF MATHEMATICS 389

to have been made at least 2000 years B. C. They show a knowledge of
numbers which indicates that their civilization must have been far
removed from the low stages in which many native tribes exist at the
present day. Simple counting with the fingers of the two hands can
be considered as the first stage, but beyond ten some new system must
be devised. It appears that the Babylonians had learned the method
of position, that is, that the first figure to the right shall represent the
units, the next figure the tens and so on. They had even constructed
numbers with 60 as the base of the system instead of ten. They could
write numbers exceeding a million, one tablet giving a table of squares,
and they also used fractions. Their geometry, however, was only in
an elementary stage. But in astronomy they seem to have passed be-
yond the first stage of observation and to have been able to classify the
results, for they possessed a knowledge of the Saros or period of
18, 2/3 years in which the eclipses of the sun and moon recur; this
must have involved a long period of observation and record as well as
the ability to classify the results and it may perhaps be regarded as
the earliest recorded. scientific deduction. from observation.

Concurrently with this civilization was one of perhaps equal de-
velopment in Egypt. The Ahmer Papyrus, which is usually dated
some 2000 years B. C., shows that the Egyptians had not only already
constructed an arithmetic but had started the solution of what we now
call equations of the first degree with one unknown. There was no
general method for solving the problems and no symbolism for the
unknown, although symbols for addition, subtraction and equality
occur. In geometry they were also somewhat in advance of the
Babylonians, apparently on account of practical needs in land survey-
ing. It is generally agreed that the pyramids show evidence of
astronomical observation in their orientation with respect to the stars
and they certainly show evidence of a knowledge of geometrical form.
As these monuments appear to go back some 4000 years B. C., we have
evidence of some considerable development much further back even
than the times indicated by the Babylonian tablets or the Ahmer
Papyrus.

But the first real evidence of the scientific spirit comes from Greece.
While they probably inherited some of the accumulated knowledge of
both Babylonia and Egypt, they transformed much of the raw material
thus acquired into a finished product which has survived to our own
times. Much the most remarkable of all that we inherit from them
in science, is the change which they effected in the study of geometry.
The love and knowledge of form which is so strikingly exhibited in
their buildings and statuary was also applied to geometrical figures.
Development of logic and a real desire to know the sources of all
things, was applied to the same study. Thus Greek geometry was not

390 THE SCIENTIFIC MONTHLY

only a consideration of the properties of straight lines and circles, but
much more, a development of those properties by processes of thought
from axioms laid down as a basis. If the modern mathematician can
see defects of logic in much that has been handed down, he still
follows Greece in the method of thought by which he deduces one
result from another. Those same methods, indeed, are used to criticize
the defects of the early work and that he is able to do so is chiefly due
to the extended range of ideas which the developments of all forms of
science have been able to give him. Some of the great names of an-
tiquity in what we now term the humanities, are also great names in
the history of science; Plato and Aristotle played no small part in the
early developments.

The Greek school appears to have started about the seventh century
before Christ, and it lasted nearly a thousand years, coming to an end
finally after a long struggle against the stifling effect of the Roman
conquest. For the first three hundred years Greece, a free nation, or at
least under the government of its own citizens, found leisure to devote
to intellectual pursuits, as a host of great names testifies. The invasion
and repulse of Asia in the fifth century B. C. seems to have quickened
rather than retarded the development of literature, science and the
arts, perhaps under the spur of the construction of a single nation with
democratic ideals from numerous small states. The conquest of Greece
by Alexander in 330 B. C. transferred the sceptre nominally to Egypt
through the enlightened policy of the Ptolemys who founded and en-
couraged schools of learning in Alexandria. But those who taught
and worked there were all under Greek inspiration, and most of the
advances were made by those of Greek origin. Apparently the foreign
soil and alien patronage, however, gave only a temporary lease of life,
for a decline started soon after and was accentuated in the first century
B. C. when the Romans conquered Egypt and gained command of the
whole civilized world. They themselves contributed little. The Fall
of the Roman Empire through the incursion of the northern tribes, the
rise and spread of Mohammed’s followers, who, it is true, brought in
other sources of knowledge from the eastern world but contributed
little themselves, the domination and repression of free thought by
ecclesiastics, left no opportunity for schools of science to grow. But
few names appear in this period and those who sought to study the
problems of nature had little opportunity to extend the borders of
knowledge.

The first signs of awakening appeared towards the end of the
fifteenth century. Revolts against ecclesiastical authority in England
by the reigning sovereign, and later in Germany by Luther, the great
geographical enterprises which soon resulted in a knowledge of the
principal land areas of the earth, the curbing of royal ambitions by

THE HISTORY OF MATHEMATICS 391

the defeat of the Spanish Armada, and the brief interlude of a com-
monwealth in England, had their counterpart in the invention of print-
ing and the appearance of scholars advancing knowledge in almost all
the civilized countries of the western world. The tremendous steps
taken in the course of two centuries, culminating in Newton and
Leibnitz, finally placed scientific investigation on a plane where it
became largely indifferent to what was going on in the political world,
and where it was able to pursue its own course, hampered it is true,
by wars and revolutions, but never without great names which will live
as long as the history of scientific development is remembered.

This rapid sketch may serve to assist in seeing how the progress of
scientific thought has been related to the development of civilization.
We are all apt to regard our own concerns as independent developments
and too often the history of science is treated in the same way. But
in looking at the subject over long periods of time, we should treat
scientific development as one of the phenomena which will illustrate
progress or decline. The attitude of mind which leads to a search for
the secrets of nature is simply one manifestation of a common desire
for progress and if so, we should see signs of this desire in many di-
rections. The ultimate causes which underlie the changes which have
taken place—which produce periods of activity and inactivity in a
whole people or group of peoples—are unknown and will probably
only be finally found in the laboratories of the biologist and the
physicist. All we can do now is to correlate the facts as far as possible
and record them for future use.

Let us return to the Greeks and examine a little more in detail their
contributions. In doing so we are at once faced by the difficulty that
most of our knowledge comes to us second hand and in the form of
treatises which gathered together past achievements. But few names
survive and it is not always easy to apportion the credit. It sometimes
appears that a name represents a school rather than an individual.
This is certainly true to some degree of Pythagoras, to whom is usually
credited the theorem that the sum of the squares on the sides of a right-
angled triangle is equal to that on the hypotenuse. He certainly formed
schools, but these were conducted as secret societies, the members of
which might not divulge the knowledge they attained. Their continued
existence for a long period of years was probably much assisted by the
custom or rule of attributing all discoveries made by the members to
their founder, thus avoiding much heartburning and jealousy. Never-
theless Pythagoras seems to have been responsible for placing
geometry on a scientific basis by investigating the theorems abstractly.
Briefly stated, it may be said that up to the middle of the fifth century
B. C., shortly after the battle of Salamis, the Greeks had discovered the
chief properties of areas in a plane and most of the regular solids.

392 THE SCIENTIFIC MONTHLY

Their clumsy notation for numbers handicapped them in arithmetic,
but they knew such properties as that the difference of the squares of
two consecutive numbers is always an odd number, they defined pro-
eressions of different kinds, and they had some acquaintance with irra-
tional numbers. This was perhaps the first school which devoted itself
to investigation for its own sake without any special reference to pos-
sible applications to physical problems.

Then followed the golden age of Greece with Hippocrates of Chios,
Euclid, Archimedes, Appolonius, Hipparchus, and a number of others
less well known. Plato and Aristotle must also be included for their
contributions to the forms of logical reasoning which should be
adopted, and Plato in particular contributed in this respect to an extent
which has lasted until our own times. Euclid, whose personality is de-
cidedly nebulous, wrote a text book on the geometry of the line and
circle by which most mathematicians for the next two thousand years
were introduced to the subject: it is indeed only during the last few
decades that it has been replaced by modern texts. Appolonius did
much the same thing for the curves of the second degree—the ellipse,
parabola and hyperbola. While we know little of Euclid’s own con-
tributions as a discoverer, it is fairly certain that Appolonius had not
only mastered all that was previously known, but greatly extended that
knowledge himself. But of all those who lived up to the time of Isaac
Newton, there can be little doubt that Archimedes is the chief. He is
recognized as the founder of mechanics, theoretical and practical. His
work on the lever alone entitles him to fame: “give me a fulcrum
and I will move the world.” He initiated the sound study of hydro-
statics, advanced geometry by discovering how to find the area of a
sphere and that cut off from a parabola by a straight line. He also dis-
cussed spiral curves, finding many of their properties. In arithmetic
he seems to have had methods for dealing with very great numbers
similar to those we now use by the index of the power of ten which
can represent the number. These brief statements are only smbolic of
what he achieved. His activities were very extended, and he preferred
the modern custom of writing essays, or memoirs as we now call them,
rather than treatises or text books. The tradition runs that he was
killed at the fall of Syracuse to the Romans because, absorbed in a
geometrical diagram, he insulted the Roman soldier who was spoiling
it.

Hipparchus was mainly an astronomer, and indeed one of the
founders of the art, and it was in the course of his work that he initiated
Trigonometry as an aid when angles had to be measured as well as
lines.

During the following centuries up to the. Fall of Rome in A. D.
476, there were no great advances, the principal name of the period

THE HISTORY OF MATHEMATICS 393

being that of Diophantus, whose main contribution was a study of in-
determinate equations. Algebra seems to have been developing very
slowly and naturally, first by abbreviation of the words of a statement,
next by a typical word (heap) and later by a symbol to represent the
unknown, and finally by the adoption of symbols for common opera-
tions, like those of addition, equality, and so on. Diophantus took a
great step in this direction, and he may be regarded as the father of
Algebra in the main stream of the development of mathematics, in the
sense that he placed it on a basis which rendered it capable of develop-
ment. However, his work received no recognition at the time and was
not revived for more than a thousand years.

We must now say something about some of the tributaries which
paralleled the main Grecian stream and connected with it through the
Moorish invasion of Europe during the 7th and 8th centuries. During
the time of their dominance the Romans, if they contributed little, at
least allowed development to continue in the centers of civilization.
The Arabs, however, following the example set by the theologians, who
by this time were beginning to come into power, had little use for
scientific works and investigation, and between them they destroyed
completely the greatest library and museum of antiquity, that of Alex-
andria, so that most of the written learning of that and of previous
ages was lost for all time. They did however form a valuable connec-
tion with the Hindu mathematics of their time and appeared not only
to have brought it to Europe, but to have assimilated it themselves to
some considerable extent.

The origin of Hindu mathematics seems very uncertain. Apparently
some centuries before our era, they had developed arithmetic further
than the Grecians of the same period, but we do not know to what ex-
tent the Persian conquest might have been responsible for introducing
early Greek learning into India. It is however true, that from what-
ever source they started, they developed arithmetic and geometry from
the mensuration point of view further than the Greeks who had con-
sidered geometry from an abstract standpoint. In illustration of this,
it may be mentioned that they had found good approximations to
V2 and to z. But their greatest contribution is our present number
system, which came to Europe through the Arabs and hence got its
name. From the seventh century it does not appear that they formed
an independent school. Hindu mathematics seems to have been largely
improved by the needs of astronomy. Their greatest exponents were
Brahmagupta, who lived in the seventh century A. D., and Bhaskara,
some five centuries later. Their work, again, is chiefly arithmetical.
While the so-called arabic numbers were probably used by the former,
the latter was the first who is known to have given a systematic treat-
ment of the decimal system. Although the Arabs dispersed the western

394 THE SCIENTIFIC MONTHLY

schools, they set up schools of their own which developed mainly on
algebraical lines. One name stands out prominently, that of Alkarismi
—who may be regarded perhaps as the founder of modern algebra—
the name of the subject itself comes from him. But he used no com-
plete symbolism, and he and his successors developed it mainly from
the arithmetical point of view. In this school the modern names of the
trigonometrical functions appeared although little was done in the way
of development.

Chinese science seems to have started about the same time as that
of Greece, but to have had little or no connection with our western
science until the sixteenth century. Whatever mathematics the Japanese
had probably came from China. It would appear at first sight that the
earliest developments arose independently in all civilized nations about
the same time, speaking broadly, but our knowledge is so scanty that
we can only say that the records nearly all date back to similar periods
in each case. Whether there is anything significant in this fact must be
left to conjecture.

With this brief sketch this era may be thought of as closed in so
far as the development of science and particularly mathematical science
is concerned. It witnesses a real beginning in the study of geometry
and algebra, and to a much less extent, of physical principles. A
system of logical reasoning was discovered which, in its main outlines,
still forms the basis of all deduction at the present day. The properties
of the simpler geometrical figures had been studied and committed to
writing. The advance of arithmetic was hindered by a poor numerical
notation, but the foundations were laid for future development by the
introduction of the Hindu symbolism. Algebra had started to emerge
from the rhetorical form of discussion into the more terse abbreviations
which we now use. Astronomy never failed to have exponents, though
many who doubtless desired to increase their knowledge were restrained
by the superstition of the age and by ecclesiastical authority, which
attempted to dictate the thoughts as well as the actions of men. In
mechanics, Archimedes seems to have stood almost alone, largely, per-
haps, because no scientific method in dealing with the fundamental
problems of nature had yet become current amongst the learned men.

It is dificult to sum up in a phrase the reasons for the scientific
hiatus which occurred during the following three or four centuries.
The revival of learning which sprang up towards the close of the eighth
century was almost barren of progress in mathematics. The schools
established by Charlemagne, while teaching mathematics, developed in-
terest in other directions. Reverence for authority appears to be the
basis of nearly all the learning of the age and there is no more stifling
attitude of mind for progressive evolution of ideas. We may attribute

THE HISTORY OF MATHEMATICS 395

this to a variety of causes any one of which may seem sufhicient to ex-
plain it, but in reality, not one of them explains anything. It may per-
haps be best likened to one of those pauses which nature seems to de-
mand and in which breath is sought before taking the next forward step.

The first sign of activity found expression in the Crusades under-
taken to free Jerusalem from Mohammedan control. Soon after this
time, that is, at the beginning of the twelfth century, several of the
modern universities were founded, partly as gatherings of students to
learn and discuss, and partly as developments of schools which had
been under the charge of the monasteries. The old Greek works on
mathematics were revived and the learning transmitted by the Arabs
began to be assimilated and taught. Systematic instruction and the
writing of treatises for the transmission of knowledge were begun.
These were mingled with courses on astrology, alchemy, and magic,
perhaps, we may conjecture, because the learned man had little chance
to earn his living except through supplying what was in common de-
mand, and perhaps also because the ancient learning gave little enough
scope for cultivation of the imagination.

A century later Roger Bacon appears on the scene. It is difficult
to overestimate the man who, travelling over Europe and studying in
Paris and doubtless absorbing the learning of his time, becoming a
Doctor in Theology and probably a monk, could break away from his
training and absolutely reject the ideas and methods of his age. One
who could write that in order to learn the secrets of nature we must first
observe, that in order to predict the future we must know the past, must
certainly have had unusual clarity of vision. He taught too that mathe-
matics was the basis of all science, but he clearly recognized that it
could not replace experiment and knowledge of phenomena, but could
be used to great advantage in deducing results when the phenomena had
once been observed: the point of view is thoroughly modern. But his
three great works failed to have much influence on his times and seem
to have been forgotten for several hundred years. Like many of his
predecessors and followers in science, he suffered for his opinions.

The real start of modern science opens, as has been mentioned
earlier, in the middle of the fifteenth century, and from this time on
there is no break in the sequence of scientific discovery. We have also
less need to relate its progress to that of the social order. While there
have been periods in which wars have seriously disturbed the ability
of nations to produce and to foster learning, there has been no time in
the period in which the whole of the civilized world was so far involved
in struggles that intellectual progress came to a stop everywhere.
Moreover, all attempts since 1450 to enslave the world, either physically
or spiritually, have ended in failure. It is true that five hundred years
is comparable with the periods of Greek, Roman, and Mohammedan

396 THE SCIENTIFIC MONTHLY

supremacy, but in each of these civilizations we see signs of decline in
a far shorter period. At present there appears to be no indication that
the crest of the wave of scientific progress has been reached. The world
war just concluded, in its essence a struggle for liberty of thought and
action, was far too short to submerge the knowledge of the existing
generation and prevent it from being handed on to the next.

The period can be roughly divided into two parts, the two centuries
until the time of Newton forming the first of these. It is chiefly
characterised by a sustained effort to lay solid foundations for all those
sciences in which mathematics is needed for development, and success
very nearly complete was attained at the end of the seventeenth century.
In Astronomy and mechanics, Copernicus placed the sun in the center
of our solar system. Kepler, building on the observations of Tycho
Brahe, gave the laws of the planetary motions. Galileo laid founda-
tions ‘for the study of mechanics and proved the rotation of the earth
about its axis, besides making a host of astronomical discoveries with
his telescope. In mathematics, the arabic numerals came into full
use and were applied to the needs of daily life. The symbolism of
algebra was completed in nearly the modern form used in elementary
mathematics: much of the work done by Vieta, Cardan and Tartaglia
involved the solution of equations of the third and fourth degrees.
Trigonometry, needed by the astronomer, the map-maker, and the navi-
gator, was developed so far that Rheticus calculated a table of natural
sines to every 10 seconds of arc and to 15 places of decimals.
Logarithms were invented by Napier, and seem to have been adopted
universally almost immediately. Descartes invented analytical geom-
etry and thus gave to the investigators of space-forms a new weapon
of immense power, while Desargues laid the foundation of projective
geometry. Fermat, an amateur and perhaps as able as any of his con-
temporaries, laid down many of the laws of numbers and founded the
calculus of probabilities.

The brevity of this summary is not a measure of the achievements
of these two centuries- For this we must look to those which followed.
While judged by modern standards, the actual progress seems small,
it was far beyond that which had been achieved in all the previous ages.
The only earlier contribution which has stood the test of time is per-
haps the geometry of the Greeks, for the work of Archimedes, especially
in mechanics, seems to have remained almost unknown up to the time
of Galileo. The period showed not only a sound laying of founda-
tions, but in its development of ideas, often only dimly expressed,
showed that the germination of the seeds of future knowledge had
already begun. Progress was frequently hampered by an unfortunate
form of rivalry in which a discovery was kept back so that its possessor
could propose problems to confute the claims for knowledge of his

THE HISTORY OF MATHEMATICS 397

contemporaries. On the other hand, the circulation of knowledge was
greatly increased by the possibility of printing old and new work.
Books became regular articles of merchandise and could even be picked
up in out of the way places.

The era of Newton is so important in the history of both pure and
applied mathematics that no excuse is necessary for dwelling on what
was achieved. If an attempt be made to characterize its results in a
single sentence, it may be perhaps best emphasized as the epoch of the
discovery of the fundamental laws of continuously varying magnitudes.
Before this time such solutions of dynamical problems as had been
obtained were isolated. Newton’s formulation of the laws of motion
and proof of the law of gravitation were found in his hands and in
those of his successors sufficient to deal with all the problems of physics
which were then and later under consideration. Little progress, how-
_ ever, could have been made without the necessary complement, the
calculus, which permitted of the study of varying quantities by sym-
bolic methods. Rates of change, when uniform, presented little
difficulty; the real problem was to deal with them when they were
variable, as, for example, in the motion of a pendulum or the vibrations
of a string. To a limited degree, the geometry of the straight line and
the circle can deal with these questions as Newton showed in the classic
translation into geometry of his results for the motions of the moon.
But his manuscripts indicate that he failed beyond a certain point to
give geometrical proofs of other. results obtained by means of the
ealculus. :

But as I have emphasized before we must not only look to the
applications, the chief question in Newton’s time, but also point out
that varying magnitudes have been studied for their own properties so
extensively that they form the larger part of mathematical develop-
ments up to the present day. If we except the science of discrete num-
bers, it is only in comparatively modern times that discontinuous
magnitudes have received any extended study and even these have been
advanced in many cases through developments of the calculus.

Isaac Newton seems to have been one of those very rare cases of
genius breaking out without any very obvious stimulus from a par-
ticular teacher or school. His first introduction to mathematics was
accidental: a book in astrology picked up at a fair in 1661 containing
geometry and trigonometry induced him to study Euclid and then to
continue by reading such text books as were available. His discovery
of the calculus or “fluxions” as he called it, was made within three
years and a half, the binominal theorem was formulated about the same
time, and a year later he began his first attempts to prove the gravita-
tion law. He was elected Professor of Mathematics in 1669, eight years

398 THE SCIENTIFIC MONTHLY

after his entry into Cambridge as an undergraduate. At this time his
chief subjects for lectures and investigation were optics and algebra,
the former of which involved him in much controversy. In fact all
through his active period of work in mathematics, he seems to have
suffered from the difficulties of having his great advances understood
and accepted, although there never seems to have been much question
amongst his contemporaries as to his wonderful powers. In 1679 he
discovered the law of areas and showed that a conic would be described
by a particle moving round a center of force under an attraction which
varied inversely as the square of the distance. But it was not until
1684 that he began to work seriously at gravitation problems, his first
step being to show that a uniform sphere exerts the same attraction as
a particle of the same mass placed at its center. From this time on
progress was rapid. Within two years the manuscript of the Principia
was finished and the following year printed.

The Principia, like most of the works of the time consists partly of
results previously known, but by far the larger part of it is Newton’s
own work. It begins with definitions, the formulation of the three
laws of motion and the principal properties which can be deduced from
them, with some examples, forming an introduction to the first book
which contains his main work on gravitation. The second book is
chiefly devoted to hydrodynamics and motion in a resisting medium
and the third to various applications of the first book to bodies and
motions in the solar system. As stated earlier, the proofs are cast into
a geometrical form and freed from all traces of the method of fluxions
which Newton had used to reach many of his results.

This great effort seems to have nearly exhausted his great powers
for he produced little after its completion. In 1695 he accepted an
appointment at the Mint and six years later resigned his chair at Cam-
bridge. He was only forty-five when the Principia was published and
he lived for thirty-eight years afterwards.

The half-century which followed the publication of the Principia
seems to have been mainly occupied in understanding and digesting
the advances made. On the continent, where Leibnitz’ notation for the
calculus was mainly adopted, a firm foundation was laid for the prog-
ress which began in the middle of the eighteenth century. In England
the reverence for Newton was so great and separation from the con-
-tinent so effective, that his methods dominated all the work for nearly
a century and a half. This was perhaps mainly due to the translation
of the Principia into geometry which ensured its early acceptance but
must have fostered a distrust of all results which could not be proved
in the same way. And further, Newton’s notation for fluxions, which
did not bring out their essential properties, had great influence in pre-
venting advances as against that invented by Leibnitz. Nevertheless

THE HISTORY OF MATHEMATICS 399

there are certain names in the English school which have lived to the
present day. Brook Taylor who was born in 1685 gave the funda-
mental series for the expansion of a function which is known by his
name, followed a little later by Colin Maclaurin with the particular
case which is named after him. The latter also determined the attrac-
tion of an ellipsoid and introduced the idea of equi-potential surfaces.
De Moivre, of French ancestry but English birth and training, intro-
duced the use of the imaginary into Trigonometry and thus prepared
the way for the great development of the theory of functions of a com-
plex variable which took place in the nineteenth century. Roger Cotes
must also be mentioned if only for the high opinion Newton had of
his abilities: he was only thirty-four years old when he died.

On the continent the Bernouilli brothers, friends and admirers of
Leibnitz, were largely responsible for realising the power of the
calculus and making it known. They applied it to many physical
problems but perhaps their greatest influence came through their teach-
ing abilities. Most of those in this period who achieved distinction
were their pupils and several of their descendents were well known as
mathematicians during the eighteenth century. But the most able men
of this period were undoubtedly Clairaut and d’Alembert. The former
produced the first theory of the motion of the moon developed from
the differential equations of motion: in it he showed that the theoretical
motion of its perigee, which in the Principia was obtained to only half
the observed value, agreed with observation when we proceed to a
higher approximation. There was an interesting development in this
connection. Clairaut at first thought that it would be necessary to make
an addition to the Newtonian law in order to produce agreement and
it was only when he carried his work further that he saw such an
addition to be unneccessary. A similar addition was examined by
Newcomb and others as a possibility which might explain the deviation
of the perihelion of Mercury from its observed value, and it is only
within the last five years that such an addition has been deduced from
the relativity theory by Einstein. Clairaut also obtained the well
known formula for the variation of gravity due to the shape of the
earth.

D’Alembert is best known by his work on dynamics in which he
showed how the equations of motion of a rigid body could be written
down: the principle is still known by his name. He also reached the
well known wave-equation, a partial differential one of the second
order, in several physical investigations and showed how a solution
may be obtained. This was pioneer work but its further development
was not carried forward to any extent by him.

The great period of continental activity which began in the middle
of the eighteenth century and contained the names of Euler, Lagrange,

400 THE SCIENTIFIC MONTHLY

Laplace, and many others of whom a brief mention only can be made,
was characterised also by social ferment which has profoundly changed
the basis of society. The French revolution with all its far reaching
consequences took place in the middle of the time of greatest mathe-
matical activity, and the Napoleonic wars, carried into almost every
country of Europe, served to stir up intercourse between the nations
to a degree which must have had much effect on all forms of scientific
activity. At the same time, the American revolution produced a new
body of political thought which has had its development in the forma-
tion of a great and powerful nation. In England, comparatively free
from invasion or social disturbance, little was produced of permanent
value, the influence of Maclaurin being directed to the retention of
published Newtonian methods. With the single exception of the
Italian Lagrange, the great names of the period belong to France and
Switzerland.

Those who have had the most far reaching influence, judged by
modern standards, besides Euler, Lagrange, and Laplace, are Legendre,
Fourier, Poisson, Monge, and Poncelet. I omit those whose chief
labors were more closely associated with experimental work. The
means for publication in this period were becoming more extensive and
it is less easy fo discover what each man owed to personal meetings
with his contemporaries. The principal academies of Europe were
starting or had started their volumes of transactions, extended treatises
could be published and circulated, and the scientific men had new op-
portunities to meet and discuss, even to some extent with those of other
countries, especially on the continent. The more enlightened courts
sought the services of the ablest scientific men and, when the latter did
not mingle too much in politics, on the whole treated them well.
Fortunately, at least for the leaders, their teaching was usually con-
fined to small bodies of earnest students and they appear to have been
little hampered by demands on their time and energy for administrative
duties. It is interesting to note that in a time of political disturbance
which was perhaps comparable to that produced by the great war, the
main foundations of modern mathematics, both pure and applied
mathematics, were firmly laid.

Leonard Euler, born in 1707 at Basle in Switzerland and educated
in mathematics by Bernouilli, was perhaps the most industrious of all
his contemporaries and it is difficult to say whether his services to
mathematics are to be judged best by his excellent treatises on analysis,
including the calculus, or by his original memoirs on applied mathe-
matics. In the former, he followed up all that was known at the time,
recasting proofs and setting the whole in logical order. In the latter,
he is best known for his formulation of the equations of motion of a
rigid body, for a similar service in the equations of motion of a fluid

THE HISTORY OF MATHEMATICS 401

and for his work on the theory of the moon’s motion. With respect to
the last it may be said that every modern method of treatment can be
found to have started with Euler. He continued his work to the end of
his life in 1783 in spite of losing his sight some fourteen years earlier
and his papers by a fire in 1777.

J. L. Lagrange was born at Turin in 1736 and was, like Newton,
practically self-educated as far as his mathematical studies were con-
cerned. It gives some insight into the comparatively small body of
mathematical literature at the time he was seventeen years old and his
own great ability, that an accident directed his taste for mathematics
and that after two years work he was able to solve a problem in the
calculus of variations which had been under discussion for half a
century. At the age of twenty-five his published work showed that in
ability he had no rival. Before his death, at the age of seventy-seven,
he had left his influence on almost every department of pure and ap-
plied mathematics. His generalizations of the equations of mechanics
have proved to be fundamental in all modern investigations and his
applications to dynamical theory of the principle of virtual work and
of the calculus of variations are now even more important than at any
time in the past. The latter is applied not only to particles and rigid
bodies, but also to fluids. To celestial mechanics he contributed several
new methods in both the theoretical and practical sides of the subject
in which such topics as the general problem of three bodies, stability
of a planetary system, mechanical quadratures and interpolation are
developed. In pure mathematics his lectures on the theory of analytic
functions, afterwards expanded in treatises, form the basis on which
later writers built. He also founded the science of differential equa-
tions by considering them as a whole rather than a treatment of such
special equations as might arise in particular problems. And he con-
tributed some important memoirs on the theory of numbers. His in-
fluence was undoubtedly much increased by a remarkable gift of ex-
position both in lecturing and writing: those who have read his
Mécanique Analytique in which his most important dynamical con-
tributions were placed will appreciate this fact. And this may be said
independently of the simpler problem before him than has the modern
mathematician with the enormous mass of past work which he has to
consider and the selection which must necessarily be made.

P. S. Laplace, whose mathematical ability is unquestioned, was
essentially an applied mathematician in that he devoted himself almost
entirely to the solution of the problems of nature by mathematical
methods. In general, he was not particular about mathematical proofs
or logic, provided he could obtain results: in many respects he may be
said to be the founder of the more modern schools of mathematical
physicists. His most enduring work has proved to be in the theory of

VOL. XII.—26.

402 THE SCIENTIFIC MONTHLY

attraction, especially gravitational, and its application to the solar
system. He made potential a real and valuable instrument of analysis
and discovery, working out many of its properties and applying it in
various directions. The famous second order differential equation
vy?V=0, which is satisfied by every gravitational arrangement of
matter, has been used as a substitute for the simpler expression of the
Newtonian law of attraction and is especially interesting at the present
time, since it may be regarded as the Newtonian analogue of the
Einstein law. Laplace was the first to attempt a complete explanation
of the motions of the bodies of the solar system or at least to formulate
methods which could be applied for this purpose and his Mécanique
Céleste remained the standard work of reference in this subject for a
century. The theory of the development of the solar system from a
primeval nebula which goes by his name and which was independently
set forth somewhat earlier by Kant, has never been entirely rejected,
although its supporters have often changed it almost beyond recognition
while retaining his name in connection with their work. Its funda-
mental idea consists of the contraction of matter under gravitational
attraction, but few now believe that the matter can produce planets
and satellites by the throwing off of concentric rings as he supposed.
What he did for celestial mechanics, he also achieved for the subject of
probability, his Théorie Analytique des Probabilités being the classic
treatise in which the whole is put on a sound basis and developed in
various directions. It must be added that he gave many theorems and
results in pure analysis but in most cases these were invented to solve
physical problems.

Of the remaining names in this period, Legendre was essentially
an analyst, his work being mainly in the theory of numbers, which
few mathematicians of this time altogether left alone, in integral
calculus and ellipitic functions, his treatises on these subjects being
still consulted. Monge and Poncelet may be regarded as the founders
of modern descriptive and projective geometry respectively. Fourier
in his Théorie Analytique de la Chaleur enunciated the theorem for
the expansion of a function in a periodic form which has had such
immense value in the discussion of all periodic phenomena, and has
now a literature of its own. Poisson’s work in attraction is on similar
lines to that of Laplace, whose natural successor he seems to be.

It is conventient to view the progress made in the nineteenth and
twentieth centuries from two points of view: one, the development of
the three great branches of mathematics, geometry, analysis and their
applications to other studies; the other, the development of new ideas
which have applications in many branches of mathematics. While it
may be said that the former is more particularly characteristic of the

THE HISTORY OF MATHEMATICS 403

first half of the nineteenth century and the latter of the succeeding
period, it would give a false idea of the method of progress to regard
these as anything more than general tendencies. But the difficulty (men-
tioned earlier,) of conveying an understanding of the advances made
in pure mathematics, even by one much more familiar with them than
the writer, occurs in an exaggerated form in attempting a chronicle
of the work of the past century. The task is far easier in applied
mathematics because most of us have some acquaintance with the
problems, and the ideas to be conveyed are not so far away from our
every day experience. Consequently, in the former, I can do little
more than point to a sign post here and there, occasionally indicate the
course of the road which has been followed, or describe by an
analogy or an example a result which has been obtained.

The older geometry which consisted in the investigation of figures
which could be generated under some simple descriptive definition like
lines, circles, or conic sections, was greatly extended by Descartes’
invention of analytical geometry. In the latter an algebraic statement
of the properties gave rise to various classes of curves which could be
ordered according to the forms of algebraic statement. Their proper-
ties could be investigated with much greater ease. The way was opened
also for the consideration of the different kinds of curves or surfaces
which possessed some definite general property; the properties com-
mon to a given class of curves or surfaces; the relations which may
exist between theorems in analysis and geometry; and so on. When
the methods of the calculus were added, the range of investigation was
again widely extended through its facility for dealing with the proper-
ties of tangents and curvature. The new results obtained were un-
doubtedly instrumental in stimulating investigation from the more
purely geometrical point of view. The names of Desargues, Monge,
and Poncelet have been already mentioned as the creators of the sub-
ject of projective geometry; their work was continued and, during the
third decade of the ninetenth century, developed into a separate branch
by Moebius, Pliicker, Steiner and a host of writers who have followed
them. Simple illustrations of the idea involved, are that of map-
making in which we represent portions of the spherical surface of the
earth on a plane, or that of the shadow of a figure cast by a ray of
light. These simple ideas have been generalised by considering the
common properties of figures which are projected according to some
given law and more generally by correspondences between two or
more figures. Another development is that of transformations which
leave properties unchanged. It will be seen at once that measure-
ments of actual lengths of lines or metric properties, as they are
called, are not those which would ordinarily be unchanged by projec-

404 THE SCIENTIFIC MONTHLY

tion, but this difficulty was overcome by Laguerre, Cayley and their
successors.

Differential geometry is in its essence a study of the properties of
geometrical figures by investigating the properties of small elements of
those figures. Such properties as curvature of a single curve or sur-
face, and those which depend on classes, such as the envelope of a
systems of curves, the surfaces which cut systems of surfaces at right
angles are the natural subjects of investigation under this head. In
1828 Gauss published a memoir which immensely extended the range
of this subject, by introducing new ideas which have been applied to
such topics as the deformation of surfaces under given conditions and
more particularly to those properties which remained unchanged by
deformation. The subject is closely allied to many problems in
physics.

Many futile attempts to deduce the axioms of parallels from the
other axioms laid down by Euclid led Lobatchewski and Bolyai to see
what would happen if a geometry free from this axiom were con-
structed. The results showed that it could be made quite consistent
in all its theorems, that some of the Euclidean theorems would still
hold while others would be changed or generalised. Their chief succes-
sor was Riemann who showed that all previous geometries were special
cases of a more general system. In our own time the subject has been
developed in the direction of finding the properties which are possessed
by the different geometries and also by investigation of the different
sets of axioms which can be used as a basis for constructing different
classes of geometries. In the applications to physics the most import-
ant has been perhaps the recognition that our own space is not neces-
sarily Euclidean and that we can only find out its nature by examining
properties which we are able to observe and measure.

The new developments have not prevented further research on the
older lines. Geometry is still much used as a convenient language for
the development and expression of analytical results. As seen below,
the plane is the home of the complex variable, but in the theory of
functions of this variable, it has become many-storied with ladders
reaching from one story to another. Most of our physical problems,
however, demand the use of three dimensional space and here the
complex variable is not sufficient because with our ordinary algebraic
rules, there are only two different kinds of numbers, the real and the
imaginary, which are used to deal with two different directions in a
plane. Hence the theory of vectors which deals with straight lines
in any number of dimensions and particularly three has been evolved
and is coming more and more into use on account of its brevity and
compactness. It requires, however, a new notation to be learnt and a
certain degree of familiarity with the operations which are possible.

THE HISTORY OF MATHEMATICS 405

The older theory of numbers in general dealt with integers and to
a less extent with fractions, square roots, etc., that is, numbers which
could be formed out of the integers by the ordinary operations of
arithmetic. Gauss opened the way to a more extended idea of the
meaning which might be attached to numbers by introducing those
which are the solutions of an ordinary algebraic equation of any degree,
whose coefficients are rational; such numbers are called algebraic.
They naturally introduced complex numbers and have properties such
as divisibility more extended than those which play a large part in our
ordinary number system. This soon demanded a theory of congruences
which, in their simplest form are numbers which, when divided by a
given factor (called the modulus), always leave the same remainder
(a residue). The theory of forms was another development which
arose. Since Gauss’ time the ideas have been greatly extended to many
other kinds of numbers in which special classes have special proper-
ties, and these classes are the main subjects of investigation.

The extensive development of the theory of functions of a complex
variable is perhaps the most significant achievement of the last hun-
dred years. The imaginary was always arising in such questions as the
solutions of quadratic equations and in the new branches. The next
step was to give a geometric interpretation of a complex number by
showing that it could represent in a single symbol the two co-ordinates
of a point ina plane. A function of a complex variable was therefore
a function which could take values over an area as against one of a
real variable which could only take values along a line. The majority
of functions become infinite for one or more values of the variable and
these infinities play the major part in the development of the theory.
To Cauchy belongs the honor of starting the work in the third decade
of the nineteenth century, examining such questions as the possibility
of developing such functions in series, their integrals, and the actual
existence of functions of different kinds. Closely following him,
Weierstrass and Riemann developed Cauchy’s ideas, the former by bas-
ing his arguments on a special form—a series of powers of the
variable—and the latter by using geometrical and even physical ideas
for progress. Their successors have merged these different modes of
development and have continued to investigate with success the repre-
sentation of a function under given conditions, and the limitations of
a given function. At the same time special functions, particularly those
known as elliptic, were being developed by Abel and Jacobi, and the
latter extended them in a very general manner. We have now many
groups of such functions, some of which have become of sufficient im-
portance to have whole treatises devoted to them.

The study of functions of real variables during the past half cen-
tury has been largely due to the critical spirit which has compelled

406 THE SCIENTIFIC MONTHLY

mathematicians to examine thoroughly the foundations of the calculus.
It consists of “all those finer and deeper questions relating to the
number system, the study of the curve, surface and other geometrical
notions, the peculiarities that functions present with reference to dis-
continuity, oscillation, differentiation and integration, as well as a
very extensive class of investigations whose object is the greatest pos-
sible extension of the processes, concepts, and results of the calculus.”
(J. Pierpont, Bull, Amer, Math. Soc., 1904, p. 147).

Ever since the invention of the calculus, the relations between two
or more variables which contained also their derivatives and known
as differential equations, have been continually brought before the eyes
of the mathematician by the physicist, owing to the fact that the
simplest symbolic statement of nearly all physical problems has been
in the form of one or more differential equations. For finding the
derivatives or integrals which arose in his work, definite rules were
generally available even if they demanded much calculation. But he
failed to find such rules for most of his differential equations, and in
fact they do not exist. The pure mathematicians, led by Cauchy, took
up the question from other points of view asking, in particular, the
nature of the function which is defined by a differential equation. This
is naturally an extension of the theory of functions and the methods
of the latter opened the way. But the questions are so difficult that
only a particular form, known as the linear, has made any considerable
progress; this form, however, does embrace a large number of the
functions whose properties had been examined. In our own generation
the subject has been extended by the consideration of equations in
which integrals also occur; these again are necessary in certain physical
problems.

Most of the progress which has been made in applied mathematics
will be treated in the article on Physics, but in addition to the remarks
at the close of this article some few words may be said of those
branches in the development of which mathematics plays the larger
part. The chief of these is the dynamics of a system of particles and
rigid bodies. W. R. Hamilton and C. G. J. Jacobi, in the second
quarter of the century, put the equations of motion of all such systems
into forms which not only permitted of remarkable generalisation, but
indicated new methods of integration which opened out research into
the general properties of such systems. The later work has been
mainly developments and applications of these methods. The par-
ticular branch of this subject known as celestial mechanics has been
continued on the practical side by extended theories of the motions of
the planets and the moon and on the theoretical side by investigations
into the general problem of three or more bodies. In the former
numerous writers have continued with increasing accuracy the work of

THE HISTORY OF MATHEMATICS 407

the eighteenth century: for the latter new foundations were laid by
Poincaré some thirty years ago.

Hydrodynamics has had less success in its applications. The pre-
diction of the tides, chiefly from the work of G. H. Darwin, and the
relative equilibrium of liquids in rotation by him and Poincaré have
advanced in a satisfactory manner, but the knowledge of the motions
of bodies in actual fluids is still in an elementary condition, especially
when an attempt is made to apply it to under-sea and air craft. This,
of course, refers, not to the experimental side, but to developments
from the equations of motion. An interesting, but now almost
neglected subject is that of vortex rings in a perfect fluid, the main
features of which were given by Helmholtz and Lord Kelvin, giving rise
to a hypothesis, now abandoned, that the fundamental atoms of matter
consisted of such rings. The motions of our atmosphere have so far
defied attempts at explanation on any general plan. The theory of
sound, on the other hand, in the hands of Helmholtz and Lord Rayleigh,
has been well developed. The theory of elasticity is almost entirely a
creation of the present century and has found many applications.

{

Many of the ideas which are now fundamental in mathematics
have had their origin in an attempt to advance some particular branch.
Development has proceeded to a certain stage by means of known
methods and then stops, owing perhaps to mathematical difficulties or
to a failure of those methods to cast further light on it. A new method
of attack has then been evolved, showing new roads by which it may be
explored, after a time leading to openings which enable the investi-
gator to continue on the earlier lines. These new methods have then
been seen to be applicable to various other branches, thus forming
connecting links and shedding new light. Indeed, one not infrequently
meets with a statement that all mathematics can be based on some one
of these ideas. This may be true, but progress demands that the sub-
ject be cross cut in many ways: a new country may be opened out by
one great highway, but it is only well developed by several main roads
in different directions with numerous connecting branches. I shall
take up certain of these ideas and try to indicate briefly their bearing
on various mathematical topics.

Some illustrations have already been given of the effect which a
critical examination of the logical processes used in mathematics has
produced. In geometry, the examination of Euclid’s axioms has led to
the discovery of ideas of space other than those which were current in
earlier times. In analysis, algebras have been constructed in which
some of the familiar rules have been dropped or changed. The way
was thus opened to the examination of the foundations on which mathe-
matics rests. Here the work gets close to a consideration of the mode

408 THE SCIENTIFIC MONTHLY

in which the human brain can think. But without going into this ques-
tion it is possible to indicate the general method followed at the pres-
ent time. At the outset a set of statements, usually called axioms or
postulates (there is a difference of opinion as to the exact meaning to
be attached to these words), is made. These statements may be re-
dundant but must not be contradictory: a complete system is one which
contains all the statements necessary for the object in view but which
has nothing unnecessary or redundant so that no statement or combi-
nation of statements can result in another of the set. To connect and
deduce, a system of reasoning is also required. From this it will be
seen that mathematical science has no necessary relation to natural
phenomena, but that it can be regarded as solely a product of the brain
and that its results are simply consequences which may be deduced
from ideas without external assistance. The last half century has seen
great progress made in clarifying our ideas and in the introduction of
rigorous methods of argument. It has now extended to the phenomena
of nature, particularly in the direction of a reconsideration of our
ideas of time and space and in an examination of our powers of ob-
servation, as will be illustrated below.

The idea of invariants has permeated every branch of pure and
applied mathematics. In its elementary forms it is not difficult to un-
derstand. Natural processes are subject to change but we can nearly
always find certain features of them which seem to remain the same
in the conditions under which we observe them, as for instance, mass
and energy. In geometry, the ratio of the circumference of a circle to
its diameter is constant, the ratio of the sections of any system of
straight lines cut by three parallel lines is the same, certain properties
of a system of curves remain unchanged under given conditions of de-
formation, and so on. In analysis, one of the commonest modes of in-
vestigation is to find out what expressions remain unchanged when a
specified change is made in the symbols. It has been said that all
physical investigations are fundamentally a search for the invariants of
nature. The various terms which occur in modern algebra, such as
discriminant, Jacobian, covariant, are special forms of the same funda-
mental idea.

The idea of permutations and combinations which few of us fail
to meet with in our every day experience, was chiefly developed in
earlier times from the point of view of the number of arrangements
which could be made of a set of objects under certain specified con-
ditions. The modern theory of groups is the natural successor of this
subject, but as has so often happened, the point of view and its develop-
ment have changed. If we have a set of symbols and replace one by
another according to a specified law, we can consider what changes
will leave unchanged certain combinations of those symbols, as well as

THE HISTORY OF MATHEMATICS 409

the number of such changes which can be made. In doing this we
naturally make a connection with the theory of invariants. Such a set
of changes is called a group. But a more fruitful idea has been ob-
tained by considering what combinations amongst the symbols, using
a specified law of combination, will always produce another of the
symbols, and will produce nothing else. As a simple and familiar
example, take the series of numbers 0,1,2, . . . with the law of addition.
If we add any two of these numbers we always get another of the same
series and we never get any other kind of number. The whole of the
series is called a group from this point of view. The even numbers
form a sub-group under this definition but the odd numbers do not
because the sum of two odd numbers is not an odd number. If we use
the same series, omitting the zero, with the law of multiplication instead
of that of addition, we again have a group, but now the odd numbers
form a sub-group. Again we may consider the operation of turning a
straight line through 60° in a plane about one end of the line.
There are obviously six positions of the line and in whatever one of
the six positions we start, a turn through 60° will always give one of
the other positions. The whole set constitutes a group.

The idea of a group of substitutions enabled Gabois and Abel,
about the middle of the nineteenth century, to open up the way to treat
algebraic equations of a degree higher than the fourth and in fact to
show that the methods used to solve equations of the second, third and
fourth degrees could not in general be applied to those of the fifth
and higher degrees. The quintic had long been a puzzle to mathe-
maticians, all attempts to give a general solution in terms of radicals
having failed. Later on, Sophus Lie applied the idea of groups to the
solution of differential equations and was able to indicate the nature of
the solutions in certain general classes. Before his time the methods
for finding them had been disconnected and apparently without any
common property. Another form of the group theory has been applied
with success to the investigation of curves and surfaces and it is not
too much to say that the idea has been one of the most fruitful in pro-
ducing progress. “When a problem has been exhibited in group
phraseology, the possibility of a solution of a certain character or the
exact nature of its inherent difficulties is exhibited by a study of the
group of the problem.”?

One of the most useful efforts of the nineteenth century mathe-
maticians has been in the direction of proving the possibility or im-
possibility of performing certain operations or ‘of solving certain prob-
lems, that is, in the investigation of existence theroems, as they are
often called. The squaring of the circle or the trisection of an angle are
two of the oldest of them and later arose the obtaining of a finite
numerical expression for the number e which is the base of the Napier-

410 THE SCIENTIFIC MONTHLY

ian system of logarithms and which arises in numerous mathe-
matical and physical investigations. The failure of attempts to solve
these problems finally led mathematicians to consider whether they
could not be proved to be insoluble under the conditions laid down.
Complete success has rewarded them. We now know that with the use
of the ruler and compasses alone, it is not possible to find a square
which shall be exactly equal in area to that of a given circle, nor given
any angle is it possible to construct the lines which shall divide it into
three equal parts. The number e too cannot be exactly expressed by
fractions or square roots or any other such simple numerical repre-
sentations, though it can be approximated to as closely as we wish by
decimals or in other ways. The labor of useless effort on the part of
the mathematician is thus avoided, though we shall still probably con-
tinue to hear of those who claim to have performed the impossible. In
our time it is quite usual as a part of an investigation to find included
in the construction of some new function or in a new representation of
a known function, a proof of its existence; especially in those cases
where the possibility may be called into question. In celestial me-
chanics an important part of Poincaré’s work consisted in proofs of the
existence or non-existence of different kinds of motion and of different
kinds of integrals. Indeed we have a considerable class of literature
which consists solely in demonstrating the existence of functions or
curves with little indication of the methods by which they may be con-
structed. The stimulating value of such researches in suggesting prob-
lems is often forgotten by those who, with some justice, complain of
their dullness.

It is strange in connection with existence theorems that some of the
problems, most simple in statement, are still unproved. Long ago
Fermat stated that there are no whole numbers which will satisfy the
statement that the sum of the n“ powers of two whole numbers is equal
to the n power of a third whole number, except when n is equal to 2.
This impossibility has been proved for all values of n up to 100 and
for a few beyond, but no general proof has yet been given that it is
universally true. Again, there is no general method which will enable
us to pick out the prime numbers, that is, those which are not divisible
by any other number except unity. In geometry we have the famous
four-color problem in which it is desired to prove that a map consisting
of countries of any shape and arrangement can always be painted with
four colors so that no two adjoining countries will receive the same
color. In these and similar cases, no exceptions to the statements have
been found and there exist no complete demonstrations of their pos-
sibility or impossibility. It is of course assumed that if they are true

1L,. E. Dickson, Paris International Congress, Vol. 2, p. 225.

THE HISTORY OF MATHEMATICS 411

a proof can be constructed without a change of our axioms concerning
number or space.

It will be seen from the sketchy remarks of the last few paragraphs
that at least one outstanding feature of pure mathematics during the
last century has been its emancipation from the trammels imposed by
any necessity for application to physical problems. It is, nevertheless,
necessary to say a few words about these applications, although the
major part of the story naturally comes under the history of physics.
Under the general term “applied mathematics,” are included at least
three methods of study. In the first and simplest, we translate the
physical problems into symbols and deduce the consequences we desire
by mathematical methods. The work consists, therefore, of little more
than an argument on lines laid down by the mathematician. In the
second, a study of the formulae and relations which have arisen from
physical problems is made, without any special desire to apply them to
the phenomena: as indicated above, much of the pure mathematics
arose in this way, even before it was recognized that such study was a
quite legitimate intellectual exercise. In the third, the mathematical
processes used by the applied mathematician are studied in order to
find out their limitations, the extent of their validity, what extensions
they will admit, how more general methods may be obtained, the best
manner of treatment, and so on. This is not by any means an infertile
source of progress, as may be illustrated by Poincaré’s work on the
divergent series which are used to calculate the places of the moon
and planets.

The most fundamental change in the attitude of applied mathe-
maticians has been in the recognition and working out of the conse-
quences of simple fundamental principles or laws. Foremost amongst
the latter is that known as the conservation of energy, brought into
prominence in the middle of the nineteenth century by the labors of
Helmholtz and Kelvin. It is now regarded as the chief invariant of the
universe and has been applied to every branch of physics. Owing to
the various forms which energy can take and to the fact that we are
practically compelled, in applying mathematics to a physical problem,
to deal only with some partial phase of it rather than with the whole,
we cannot always assume that the principle holds in a particular prob-
lem. But in the majority of such cases the energy which is lost or
changed from the particular form which we are considering is small
so that this loss may be neglected or allowed for. When the loss is
zero, the differential equations of the problem admit of an integral
which expresses this fact. Newtonian mechanics lead to two forms of
energy, kinetic (that due to motion) and potential (that due to posi-
tion) and the development of the mathematics of all material systems
has been mainly based on this separation.

412 THE SCIENTIFIC MONTHLY

The principle of Least Action, enunciated by Maupertius in 1744
but first put into correct mathematical form by W. R. Hamilton a
century later, is essentially one which demands mathematical treatment.
The Action of a system is a certain function depending on the velocities
and relations of its parts which takes a minimum value whenever the
system moves under natural laws. The process of discovery of this
minimum leads to the differential equations of motion of the system
and thus includes a complete statement of the problem. Since the
initial form of the function is an integral, its mathematical treatment
consists in finding the least value of this integral and thus becomes a
problem in the calculus of variations to which considerable attention
has been given by pure mathematicians, especially during the last two
or three decades. While the physical consequences of the principle
have been less developed than those of energy, there appears to be a
growing feeling as to its fundamental importance and the aid of the
mathematician in solving the problems which it raises, will become
increasingly necessary.

The study of the properties of a system containing a large number
of particles not fixed relatively to one another, now generally studied
under the term, statistical mechanics, has penetrated into several
branches. It is to be understood here that the question is not one of
finding the separate motions of the various particles but to try and
find out such properties of the system as can be deduced from averages.
It is probable that as long as we cannot observe the motions of the
separate particles, we should be able to deduce in this way most if not
all the properties of the system that we are able to observe. Maxwell
and Boltzmann founded the subject from this point of view, applying
it to the kinetic theory of gases, while J. W. Gibbs was largely respon-
sible for its application to thermodynamics. In astronomy the present
century has seen it applied to the motions and positions of the stars,
thus opening the way to a knowledge of the outlines of the construc-
tion of the stellar universe. Mathematically these questions are
obviously very similar to those parts of probability which deal with
errors of observation and thus form a continuation of the development
of that subject.

The mathematics of continuous media has received very complete
development during the century and, besides the earlier investigations
into the motions of fluids and elastic solids, has been applied to the
so-called luminiferous ether and finally by Maxwell to the whole elec-
tric field. In all this work the continuity of the medium is a funda-
mental axiom involving the hypothesis that no action can take place
without its presence. Further, the Newtonian laws of motion were
assumed as fundamental and. time and length were regarded as un-
changeable separate entities. The classic experiment of Michelson and

THE HISTORY OF MATHEMATICS 413

Morley which showed that the velocity of light was apparently inde-
pendent of the velocity of the medium in which it travelled, and obser-
vations on the motions of certain particles with very high velocities,
started a reconstruction of ideas. It was possible to explain the re-
sults on the assumption that the length of a body depended on its
velocity. It was then that Einstein sought to generalize Newton’s equa-
tions of motion by making them entirely relative, not only for uniform
velocity, but also for accelerated motion. By adding the assumption
that the laws of nature should refer to all such systems of reference
and by making the velocity of light a fundamental constant of nature,
he was finally able to generalize the whole subject. Matter appears
simply as a form of energy. Gravitation can be exhibited as due to a
warping of space without the introduction of force, but if this is so,
the Newtonian law requires a minute correction. The motion of the
perihelion of Mercury and the bending of a light ray as it passes near
the sun have given remarkable confirmation of this theory. His work
crosscuts several subjects which previously had an interest only for
the pure mathematician, in particular the theory of extensible vectors
(tensors) and the theory of invariants. His differential equations for the
gravitational field should supply mathematicians with problems of
great difficulty and interest for some time to come. Those for the elec-
tric field are unchanged. A further interesting product of this work
on relativity lies in the question of what we can or cannot observe and
in what may be deduced from observation without the assumption of
hypotheses. This is, of course, fundamental in all experiments, but it
has received little attention as an exact science. Given that we can only
observe certain properties of a function, what limitations is it possible
to make in the construction of the function?

Finally the quantum theory of Planck, according to which energy
is not infinitely divisible but is always received or emitted in exact
multiples of a fundamental unit, is bringing forward the necessity for
a calculus allied to that of finite differences as against the differential
and integral calculus which depends in general on continuity. It is
even suggested that not only energy, but also space and time have
ultimate parts which cannot be divided. At present the mechanics of
this theory is in a very nebulous state but as a statement of the results
of observation it has had very considerable success. The construction
of the atom is now generally exhibited as a kind of minute solar system,
but there is as yet no indication how such a system can only permit of
the limited number of motions required by the quantum hypothesis.
It may perhaps be due to fundamental instabilities for we know little
of the ultimate stability of most of the motions in the problems of
even three particles. In any case, the field of work has approached one
of the oldest of mechanical problems and the reaction of celestial me-
chanics on that of the atom should prove stimulating to both.